A function $f$ is Log-Lipschitz if there exists a constant $C >0$ such that \begin{equation} |f(x) - f(y)| \le C|x-y| |\log|x-y|| \end{equation}

I am trying to construct two functions with the following properties.

First function is $f\in \mathcal{C}^1((0,a])$ ,$a>0$ and log-Lipschitz continuous on $[0,a]$ such that \begin{equation} \limsup_{x \to0^+} x^q|f'(x)|=+\infty, \forall q \geq 1 \end{equation} Second function is $g\in \mathcal{C}^1((0,a])$ continuous on $[0,a]$ but Holder continuous on $[0,a]$ for no $\alpha<1$ such that \begin{equation} \limsup_{x \to0^+} x|g'(x)|<+\infty. \end{equation} I tried constructing but not getting through much(I came across these functions in context of log-lipschitz regularity of certain hyperbolic pdes). Thanks in advance.